In this appendix we discuss some details of the transformation of a
Hamiltonian with a quadratic part (34)
into generalized normal form. More
specifically, we show how to simplify the Lie operator
The unitary matrix that by a similarity transformation puts the
In the new coordinates
the Lie operator takes
on the form
We have not yet made any assumptions about the ordering of the monomials in the basis of . If one chooses the lexicographical ordering  of the basis monomials, then the matrix representation of becomes an upper diagonal matrix for all , and all the manipulations of that are necessary in the course of the normalization procedure (solving linear equations, inverting , ...) become easier and consume much less computing time.
In the case it is possible to achieve even further simplification by an appropriate ordering of the basis monomials of . One can introduce the so-called magnetic bottle ordering of monomials which results in being bidiagonal.